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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 63162ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 63162.bt2 | 63162ca1 | \([1, -1, 1, 5422, 253905]\) | \(13651919/29696\) | \(-38351432807424\) | \([]\) | \(162000\) | \(1.2907\) | \(\Gamma_0(N)\)-optimal |
| 63162.bt1 | 63162ca2 | \([1, -1, 1, -495518, -135043455]\) | \(-10418796526321/82044596\) | \(-105957967763545524\) | \([]\) | \(810000\) | \(2.0954\) |
Rank
sage: E.rank()
The elliptic curves in class 63162ca have rank \(0\).
Complex multiplication
The elliptic curves in class 63162ca do not have complex multiplication.Modular form 63162.2.a.ca
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
